With few exceptions, unless you are a really good swimmer or coming from Malaysia, you'll arrive in Singapore via Changi Airport. Though the airport boasts frighteningly efficient customs and immigration processing, take a few moments to look around. There are branches of many local popular eating places at the airport, free movies in a theatre settings, reasonably priced massages, shops and truck loads of free tourist information. Be sure to pick up maps and brochures on Singapore before leaving. You can use a handy service to arrange a hotel room if you need one. There are also money changing places with exchange rates about the same as downtown.
If you arrive in Singapore roughly during daylight hours or before around midnight, you can take the MRT subway directly into downtown from the airport for a few dollars for the whole family. It is clean, safe and fast but starts around 6 am and quits around midnight. Same for the so-called limo buses and mini-vans: similar hours, not much more expensive and they will drop you right at your bedside (OK, actually in front of your hotel).
Even if you arrive after hours, a taxi from the airport to central anywhere downtown will be around US$15-25 (as of 2004). You get the taxi from a well-marked stand just outside the arrival hall. Do watch out for touts that will try and direct you out of line to a more expensive option, and make sure the driver switches on the meter. There are surcharges for early and late trips, an airport charge, luggage charges, extra passenger charges and the like that are added on to the meter at the end, but again the total price should be still in the US$15-25 (as of 2004) zone.
Coming home, your hotel can pre-book a taxi for a small fee for you. If you are arriving or departing on a very early or very late flight, the taxi is your only available transport choice.